How to test for difference in means between 5 groups? The variance between the groups are not equal. If they were equal, then I could use ANOVA There are five groups and I want to test for difference in means between the five groups assuming the variance are not constant between the groups.
 A: One of several possible methods is to use oneway.test in R.
Here is an example with three groups, each with ten observations:
x1 = rnorm(10, 100, 10);  x2 = rnorm(10, 95, 15);  x3 = rnorm(10, 90, 5)
x = c(x1, x2, x3); group = rep(1:3, each=10)
boxplot(x ~ group)


You can see that my fake data were generated with different standard deviations in each of the three groups. The boxplots show this heteroscedasticity. A Bartlett
test confirms significance (P-value 0.004).
sd(x1); sd(x2); sd(x3)
[1] 10.88923
[1] 16.35099
[1] 4.656118

bartlett.test(x, group)

        Bartlett test of homogeneity of variances

data:  x and group
Bartlett's K-squared = 11.103, df = 2, p-value = 0.003881

The oneway.test procedure allows for different variances in somewhat the
same way as does the Welch 2-sample t test. It indicates that not all
group population means are equal (P-value < 5%). Notice that the denominator DF $\approx 15;$ a standard ANOVA assuming equal variances would have denominator DF $= 27.$ [I believe this is the 'Welch ANOVA' suggested by @SalMangiafico.]
mean(x1); mean(x2); mean(x3)
[1] 98.00458
[1] 105.3806
[1] 92.0077

oneway.test(x ~ group)

        One-way analysis of means (not assuming equal variances)

data:  x and group
F = 3.818, num df = 2.000, denom df = 14.536, p-value = 0.04649

You could use Welch 2-sample t tests to explore paired comparisons, perhaps
with a Bonferroni family error rate.
Reference: This Q & A mentions a variety of alternative methods.
